I don't know if it will get you past the login page, but it's worth a try. To my code above, add a cookie jar.

my $jar = HTTP::Cookies->new;
$ua = LWP::UserAgent->new;
$ua->cookie_jar($jar);
[download]

If that don't work, you will need WWW::Mechanize. See WWW:::Mechanize and credentials and WWW::Mechanize to Access HTTPS with Cookies for basic outlines of what you need to do.